On the product of Weak Asplund locally convex spaces

Abstract

For locally convex spaces, we systematize several known equivalent definitions of Fréchet (G\^ ateaux) Differentiability Spaces and Asplund (Weak Asplund) Spaces. As an application, we extend the classical Mazur's theorem as follows: Let $E$ be a separable Baire locally convex space and let $Y$ be the product ${\prod }_{\alpha \in A}{E}_{\alpha }$ of any family of separable Fréchet spaces; then the product $E\times Y$ is Weak Asplund. Also, we prove that the product $Y$ of any family of Banach spaces $(E_{\alpha })$ is an Asplund locally convex space if and only if each $E_{\alpha }$ is Asplund. Analogues of both results are valid under the same assumptions, if $Y$ is the $\Sigma$-product of any family $(E_{\alpha })$.